Godfrey Harold Hardy

G H Hardy's father, Isaac Hardy, was bursar and an art master at Cranleigh school. His mother Sophia had been a teacher at Lincoln Teacher's Training School. Both parents were highly intelligent with some mathematical skills but, coming from poor families, had not been able to have a university education. Hardy (he was always known as Hardy except to one or two close friends who called him Harold) attended Cranleigh school up to the age of twelve with great success [5]:-

His parents knew he was prodigiously clever, and so did he. He came top of his class in all subjects. But, as a result of coming top of his class, he had to go in front of the school to receive prizes: and that he could not bear.

Hardy did not appear to have the passion for mathematics that many mathematicians experience when young. Hardy himself writes in [4]:-

I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.

Indeed he did win a scholarship to Winchester College in 1889, entering the College the following year. Winchester was the best school in England for mathematical training yet, despite admitting later in life that he had been well-educated there, Hardy disliked everything about the school other than the academic training he received. Like all public schools it was a rough place for a frail, shy boy like Hardy. It is significant that although he did have a passion for ball games in general and cricket in particular, he was never coached in sport at Winchester. Somehow he failed to take part fully in the non-academic activities.

While at Winchester Hardy won an open scholarship to Trinity College, Cambridge, which he entered in 1896. At Cambridge Hardy was assigned to the most famous coach R R Webb. He quickly realised that the point of the training was simply to achieve the best possible marks in the examinations by learning all the tricks of the trade. He was shocked to discover that Webb was not interested in the subject of mathematics, only in the tricks of examinations.

Briefly Hardy thought he might change topics and study history instead. However, he managed to change his coach to A E H Love. Hardy expresses his gratitude to Love in [4]:-

My eyes were first opened by Professor Love, who first taught me a few terms and gave me my first serious conception of analysis. But the great debt which I owe to him was his advice to read Jordan's "Cours d'analyse"; and I shall never forget the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.

Hardy was placed as fourth wrangler in the Mathematical Tripos of 1898, a result which continued to annoy him for, despite feeling that the system was very silly, he still felt that he should have come out on top. Hardy was elected a fellow of Trinity in 1900 then, in 1901, he was awarded a Smith's prize jointly with J H Jeans 'with unspecified relative merit'.

The next period of Hardy's career was up to 1911 when, as Burkill writes in [1], he:-

... wrote many papers on the convergence of series and integrals and allied topics. Although this work established his reputation as an analyst, his greatest service to mathematics in this early period was A course of pure mathematics (1908). This work was the first rigorous English exposition of number, function, limit, and so on, adapted to the undergraduate, and thus it transformed university teaching.

This was a period of which Hardy wrote himself [4]:-

I wrote a great deal... but very little of any importance; there are not more than four of five papers which I can still remember with some satisfaction.

It is worth noting at this point that Hardy was a remarkably honest man, and in particular he was very honest about his own abilities, strengths and weaknesses.

A major change in Hardy's work came about in 1911 when he began his collaboration with J E Littlewood which was to last 35 years. Then in early 1913 he received Ramanujan's first letter from India which was to start his second major collaboration. By the time World War I started in 1914, Ramanujan was in Cambridge and this eased for Hardy what was to be a very difficult period.

Littlewood left Cambridge for war service in the Royal Artillery. Hardy volunteered for war service but was rejected on medical grounds. However Hardy's views on the war left him at odds with most of his colleagues at Cambridge. He had great respect for Germany [5]:-

... he had a strong feeling for Germany. Germany had, after all, been the great educating force of the nineteenth century. To Eastern Europe, to Russia, to the United States, it was the German universities which had taught the meaning of research. ... in most respects the German culture, including its social welfare, appeared to him higher than his own. ... Hardy, like Russell ... did not believe that the war should have been fought. Further, with his ingrained distrust of English politicians, he thought the balance of wrong was on the English side.

Deeply unhappy at Cambridge, Hardy took the opportunity to leave in 1919 when he was appointed as Savilian professor of geometry at Oxford. These were in many ways the years when he was happiest and also the years when he produced his best mathematics in the collaboration with Littlewood. This collaboration was achieved during a period when Littlewood was in Cambridge and Hardy was in Oxford, making joint research a quite difficult logistical exercise. As Hardy wrote in [4]:-

I was at my best at a little past forty, when I was a professor at Oxford.

Despite his background and the positions he held, Hardy preferred the poor and disadvantaged to those he called the 'large bottomed' who included [5]:-

... the confident, booming, imperialist bourgeois English. The designation included most bishops, headmasters, judges, and all politicians, with the single exception of Lloyd George.

He had chosen not to live in the best rooms while at Cambridge, and Hilbert was so concerned that Hardy was not being properly treated that he wrote to the Master of the College pointing out that the best mathematician in England should have the best rooms. However, Hardy did not think that way. He held a trade union office for two years (1924-26) as President of the Association of Scientific Workers. At a time when it seemed difficult to do so, Hardy liked equally both the United States and Russia. He spent the academic year 1928-29 at Princeton in an exchange with Veblen, who spent the year in Oxford.

Despite having been unhappy at Cambridge, Hardy returned to the Sadleirian chair there in 1931 when Hobson retired. Snow in [5] says that Hardy returned to Cambridge for two reasons, firstly that he still considered Cambridge the centre of English mathematics and the Sadleirian chair the foremost mathematics chair in England, and secondly, that he could keep his rooms in College at Cambridge while this was not possible at Oxford. To the unmarried Hardy, this held an attraction as he began to look toward old age.

Hardy's interests covered many topics of pure mathematics - Diophantine analysis, summation of divergent series, Fourier series, the Riemann zeta function, and the distribution of primes. His long collaboration with Littlewood produced mathematics of the highest quality. It was a collaboration in which Hardy acknowledged Littlewood's greater technical mathematical skills, but at the same time Hardy brought great talents of mathematical insight and a great ability to write their work up in papers with great clarity.

Even more remarkable was Hardy's collaboration with Ramanujan. Hardy instantly spotted Ramanujan's genius from a manuscript sent to him by Ramanujan from India in 1913. Two other top class mathematicians had previously failed to spot the genius. Hardy brought Ramanujan to Cambridge and they wrote five remarkable papers together.

It was not only with Littlewood and Ramanujan that Hardy collaborated. He was a natural collaborator who also wrote joint papers with Titchmarsh, Ingham, Edmund Landau, Pólya, E M Wright, W W Rogosinski and Marcel Riesz.

Hardy was a pure mathematician who hoped his mathematics could never be applied. However in 1908, near the beginning of his career, he gave a law describing how the proportions of dominant and recessive genetic traits would be propagated in a large population. Hardy considered it unimportant but it has proved of major importance in blood group distribution.

There was only one passion in Hardy's life other than mathematics and that was cricket. In fact for most of his life his day, at least during the cricket season, would consist of breakfast during which he read The Times studying the cricket scores with great interest. After breakfast he would work on his own mathematical researches from 9 o'clock till 1 o'clock. Then, after a light lunch, he would walk down to the university cricket ground to watch a game. In the late afternoon he would walk slowly back to his rooms in College. There he took dinner, which he followed with a glass of wine. When cricket was not in season, it was the Australian cricket scores he would read in The Times and he would play real tennis in the afternoons.

Hardy was known for his eccentricities. He could not endure having his photograph taken and only five snapshots are known to exist. He also hated mirrors and his first action on entering any hotel room was to cover any mirror with a towel. He always played an amusing game of trying to fool God (which is also rather strange since he claimed all his life not be believe in God). For example, during a trip to Denmark he sent back a postcard claiming that he had proved the Riemann hypothesis. He reasoned that God would not allow the boat to sink on the return journey and give him the same fame that Fermat had achieved with his "last theorem".

Another example of his trying to fool God was when he went to cricket matches he would take what he called his "anti-God battery". This consisted of thick sweaters, an umbrella, mathematical papers to referee, student examination scripts etc. His theory was that God would think that he expected rain to come so that he could then get on with his work. Since Hardy thought that God would then have the sun shine all day to spite him, he would be able to enjoy the cricket in perfect sunshine.

As World War I had been painful for Hardy, World War II was equally so. He had remained remarkably youthful in both mind and body until 1939 when, at the age of 62, he had a heart attack. His remarkable mental powers began to leave him and sports which he had loved to participate in up till then became impossible. He was filled with anger that Europe had again entered the lunacy of war. However, Hardy had one further gift to leave to the world, namely A mathematicians apology which has inspired many towards mathematics.

Hardy's book A mathematicians apology was written in 1940. It is one of the most vivid descriptions of how a mathematician thinks and the pleasure of mathematics. But the book is more, as Snow writes in [5]:-

A mathematicians apology is, if read with the textual attention it deserves, a book of haunting sadness. Yes, it is witty and sharp with intellectual high spirits: yes, the crystalline clarity and candour are still there: yes, it is the testament of a creative artist. But it is also, in an understated stoical fashion, a passionate lament for creative powers that used to be and that will never come again. I know nothing like it in the language: partly because most people with the literary gift to express such a lament don't come to feel it: it is very rare for a writer to realise, with the finality of truth, that he is absolutely finished.

The following quotation from A mathematicians apology ([4]) gives a clear idea of Hardy's thoughts on mathematics:-

The mathematician's pattern's, like those of the painter's or the poet's, must be beautiful, the ideas, like the colours or the words, must fit together in a harmonious way. There is no permanent place in the world for ugly mathematics.

By the time the war ended in 1945 Hardy health was failing fast. He longed to be creative again, for that was all that really mattered to him in life, but he knew that his creativity was gone and that he became very depressed. By 1946 he could only get around by taking taxi rides, a few steps would make him short of breath. In early summer of 1947 he tried to take his own life by taking a large dose of barbiturates. He took so many, however, that he was sick and survived. Snow writes [5]:-

In the Evelyn nursing home, Hardy was lying in bed. As a touch of farce, he had a black eye. Vomiting from the drugs, he had hit his head on the lavatory basin. He was self-mocking. He had made a mess of it. ...

He talked a little, nearly every time I saw him, about death. He wanted it. He didn't fear it: what was there to fear in nothingness? His hard intellectual stoicism had come back. He would not try to kill himself again. He was not good at it. He was prepared to wait. With an inconsistency which might have pained him - for he ... believed in the rational to an extent that I thought irrational - he showed an intense hypochondriac curiosity about his own symptoms.

Hardy received many honours for his work. He was elected a Fellow of the Royal Society in 1910, he received the Royal Medal of the Society in 1920 and Sylvester Medal of the Society in 1940:-

... for his important contributions to many branches of pure mathematics.

He also received the Copley Medal of the Royal Society in 1947:-

... for his distinguished part in the development of mathematical analysis in England during the last thirty years.

Hardy learnt of the award only a few weeks before his death.

He is described in [3] as follows:-

He personified the popular idea of the absent-minded professor. But those who formed the idea that he was merely an absent-minded professor would receive a shock in conversation, where he displayed amazing vitality on every subject under the sun. ... He was interested in the game of chess, but was frankly puzzled by something in its nature which seemed to come into conflict with his mathematical principles.

He was president of the London Mathematical Society from 1926 to 1928 and again from 1939 to 1941. He received the De Morgan Medal of the Society in 1929.

Article by:J J O'Connor and E F Robertson

October 2003

MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/Biographies/Hardy.html]